Research Article
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Inequalities on the geometric-arithmetic index

Year 2021, Volume: 50 Issue: 3, 778 - 794, 07.06.2021

Ana Granados , Ana Portilla , Jose Manuel Rodrıguez Garcıa , Jose Sigarreta

https://doi.org/10.15672/hujms.749744
Cited By: 1

Abstract

Although the notion of geometric-arithmetic index has been introduced in the chemical graph theory these past years, it has already proved to be useful. The objective of the work we present here is twofold: First, obtaining new relations connecting the geometric-arithmetic index with other topological indices; second, to characterize graphs which are extremal with respect to those relations.

Keywords

geometric-arithmetic index vertex-degree-based topological index variable Zagreb index

Supporting Institution

Ministerio de Economia y Competitividad, Agencia Estatal de Investigacion, Fondo Europeo de Desarrollo Regional

Project Number

MTM2016-78227-C2-1-P; MTM2017-90584-REDT

References

  • [1] H. Abdo, D. Dimitrov and I. Gutman, On extremal trees with respect to the F-index, Kuwait J. Sci. 44 (3) 1–8, 2017.
  • [2] M.O. Albertson, The irregularity of a graph, Ars Comb. 46, 219–225, 1997.
  • [3] A. Ali, I. Gutman, E. Milovanović and I. Milovanović, Sum of Powers of the Degrees of Graphs: Extremal Results and Bounds, MATCH Commun. Math. Comput. Chem. 80, 5–84, 2018.
  • [4] V. Andova and M. Petrusevski, Variable Zagreb Indices and Karamataís Inequality, MATCH Commun. Math. Comput. Chem. 65, 685–690, 2011.
  • [5] M. Aouchiche and P. Hansen, Comparing the geometric-arithmetic Index and the spectral radius of graphs, MATCH Commun. Math. Comput. Chem. 84 (2), 473–482, 2020.
  • [6] M. Aouchiche and V. Ganesan, Adjusting geometric-arithmetic index to estimate boil- ing point, MATCH Commun. Math. Comput. Chem. 84, 483–497, 2020.
  • [7] M. Aouchiche, I. El Hallaoui and P. Hansen, Geometric-Arithmetic index and mini- mum degree of connected graphs, MATCH Commun. Math. Comput. Chem. 83 (1), 179–188, 2020.
  • [8] Z. Che and Z. Chen, Lower and Upper Bounds of the Forgotten Topological Index, MATCH Commun. Math. Comput. Chem. 76, 635–648, 2016.
  • [9] Y. Chen and B. Wu On the geometric-arithmetic index of a graph, Discrete Appl. Math. 254, 268–273, 2019.
  • [10] K.C. Das, On geometric-arithmetic index of graphs, MATCH Commun. Math. Com- put. Chem. 64, 619–630, 2010.
  • [11] K.C. Das, I. Gutman and B. Furtula, On first geometric-arithmetic index of graphs, Discrete Appl. Math. 159, 2030–2037, 2011.
  • [12] K.C. Das, I. Gutman and B. Furtula, Survey on Geometric-Arithmetic Indices of Graphs, MATCH Commun. Math. Comput. Chem. 65, 595–644, 2011.
  • [13] S.S. Dragomir, A Survey On Cauchy-Bunyakovosky-Schwarz Type Discrete Inequali- ties, J. Inequal. Pure and Appl. Math. 4 (3), Art. 63, 2003.
  • [14] M. Drmota, Random Trees: An Interplay Between Combinatorics and Probability, Springer, Wien-New York, 2009.
  • [15] M. Eliasi, A. Iranmanesh and I. Gutman, Multiplicative versions of first Zagreb index, Commun. Math. Comput. Chem. 68 (1), 217–230, 2012.
  • [16] G.H. Fath-Tabar, Old and New Zagreb Indices of Graphs, MATCH Commun. Math. Comput. Chem. 65, 79–84, 2011.
  • [17] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (4), 1184–1190, 2015.
  • [18] X. Guo and Y. Gao Arithmetic-geometric spectral radius and energy of graphs, MATCH Commun. Math. Comput. Chem. 83, 651–660, 2020.
  • [19] I. Gutman, Degree–based topological indices, Croat. Chem. Acta 86, 351–361, 2013.
  • [20] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 83–92, 2004.
  • [21] I. Gutman and T. Réti, Zagreb group indices and beyond, Int. J. Chem. Model. 6 (2-3), 191–200, 2014.
  • [22] I. Gutman and J. Tošović, Testing the quality of molecular structure descriptors. Vertex–degreebased topological indices, J. Serb. Chem. Soc. 78 (6), 805–810, 2013.
  • [23] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535–538, 1972.
  • [24] J.C. Hernandez, J.M. Rodriguez and J.M. Sigarreta On the geometric-arithmetic in- dex by decompositions, J. Math. Chem. 55, 1376–1391, 2017.
  • [25] H. Kober, On the arithmetic and geometric means and on Hölderís inequality, Proc. Amer. Math. Soc. 9, 452–459, 1958.
  • [26] X. Li and H. Zhao, Trees with the first smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50, 57–62, 2004.
  • [27] X. Li and J. Zheng, A unified approach to the extremal trees for different indices, MATCH Commun. Math. Comput. Chem. 54, 195–208, 2005.
  • [28] M. Liu and B. Liu, Some properties of the first general Zagreb index, Australas. J. Combin. 47, 285–294, 2010.
  • [29] A. Martínez-Pérez, J.M. Rodríguez and J.M. Sigarreta, A new approximation to the geometric-arithmetic index, J. Math. Chem. 56, 1865–1883, 2018, DOI: 10.1007/s10910-017-0811-3.
  • [30] A. Miličević and S. Nikolić, On variable Zagreb indices, Croat. Chem. Acta 77, 97– 101, 2004.
  • [31] S. Nikolić, A. Miličević, N. Trinajstić and A. Jurić, On Use of the Variable Zagreb $^\nu M_2$ Index in QSPR: Boiling Points of Benzenoid Hydrocarbons, Molecules 9, 1208–1221, 2004.
  • [32] M. Randić, Novel graph theoretical approach to heteroatoms in QSAR, Chemometrics Intel. Lab. Syst. 10, 213–227, 1991.
  • [33] M. Randić, On computation of optimal parameters for multivariate analysis of structure-property relationship, J. Chem. Inf. Comput. Sci. 31, 970–980, 1991.
  • [34] M. Randić, D. Plavšić and N. Lerš, Variable connectivity index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 41, 657–662, 2001.
  • [35] J.M. Rodríguez, J.L. Sánchez and J.M. Sigarreta, On the first general Zagreb index, J. Math. Chem. 56, 1849–1864, 2018, DOI: 10.1007/s10910-017-0816-y.
  • [36] J.M. Rodríguez and J.M. Sigarreta, On the Geometric-Arithmetic Index, MATCH Commun. Math. Comput. Chem. 74, 103–120, 2015.
  • [37] J.M. Rodríguez and J.M. Sigarreta, Spectral properties of geometric-arithmetic index, Appl. Math. Comput. 277, 142–153, 2016.
  • [38] J.M. Rodríguez and J.M. Sigarreta, New Results on the Harmonic Index and Its Generalizations, MATCH Commun. Math. Comput. Chem. 78 (2), 387–404, 2017.
  • [39] J.M. Sigarreta, Bounds for the geometric-arithmetic index of a graph, Miskolc Math. Notes, 16 (2), 1199–1212, 2015.
  • [40] M. Singh, K.Ch. Das, S. Gupta and A.K. Madan, Refined variable Zagreb indices: highly discriminating topological descriptors for QSAR/QSPR, Int. J. Chem. Model- ing, 6 (2-3), 403–428, 2014.
  • [41] TRC Thermodynamic Tables. Hydrocarbons; Thermodynamic Research Center, The Texas A & M University System: College Station, TX, 1987.
  • [42] R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2), 359–372, 2010.
  • [43] M. Vöge, A.J. Guttmann and I. Jensen, On the number of benzenoid hydrocarbons, J. Chem. Inf. Comput. Sci. 42, 456–466, 2002.
  • [44] D. Vukičević, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta, 83, 261–273, 2010.
  • [45] D. Vukičević and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46, 1369–1376, 2009.
  • [46] D. Vukičević and M. Gašperov, Bond Additive Modeling 1. Adriatic Indices, Croat. Chem. Acta, 83 (3), 243–260, 2010.
  • [47] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69, 17–20, 1947.
  • [48] S. Zhang, W. Wang and T.C.E. Cheng, Bicyclic graphs with the first three smallest and largest values of the first general Zagreb index, MATCH Commun. Math. Comput. Chem. 55, 579–592, 2006.
  • [49] H. Zhang and S. Zhang, Unicyclic graphs with the first three smallest and largest values of the first general Zagreb index, MATCH Commun. Math. Comput. Chem. 55, 427–438, 2006.
  • [50] B. Zhou, I. Gutman and T. Aleksić, A note on Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60, 441–446, 2008.
  • [51] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47, 210–218, 2010.

Year 2021, Volume: 50 Issue: 3, 778 - 794, 07.06.2021

Ana Granados , Ana Portilla , Jose Manuel Rodrıguez Garcıa , Jose Sigarreta

https://doi.org/10.15672/hujms.749744
Cited By: 1

Abstract

Project Number

MTM2016-78227-C2-1-P; MTM2017-90584-REDT

References

  • [1] H. Abdo, D. Dimitrov and I. Gutman, On extremal trees with respect to the F-index, Kuwait J. Sci. 44 (3) 1–8, 2017.
  • [2] M.O. Albertson, The irregularity of a graph, Ars Comb. 46, 219–225, 1997.
  • [3] A. Ali, I. Gutman, E. Milovanović and I. Milovanović, Sum of Powers of the Degrees of Graphs: Extremal Results and Bounds, MATCH Commun. Math. Comput. Chem. 80, 5–84, 2018.
  • [4] V. Andova and M. Petrusevski, Variable Zagreb Indices and Karamataís Inequality, MATCH Commun. Math. Comput. Chem. 65, 685–690, 2011.
  • [5] M. Aouchiche and P. Hansen, Comparing the geometric-arithmetic Index and the spectral radius of graphs, MATCH Commun. Math. Comput. Chem. 84 (2), 473–482, 2020.
  • [6] M. Aouchiche and V. Ganesan, Adjusting geometric-arithmetic index to estimate boil- ing point, MATCH Commun. Math. Comput. Chem. 84, 483–497, 2020.
  • [7] M. Aouchiche, I. El Hallaoui and P. Hansen, Geometric-Arithmetic index and mini- mum degree of connected graphs, MATCH Commun. Math. Comput. Chem. 83 (1), 179–188, 2020.
  • [8] Z. Che and Z. Chen, Lower and Upper Bounds of the Forgotten Topological Index, MATCH Commun. Math. Comput. Chem. 76, 635–648, 2016.
  • [9] Y. Chen and B. Wu On the geometric-arithmetic index of a graph, Discrete Appl. Math. 254, 268–273, 2019.
  • [10] K.C. Das, On geometric-arithmetic index of graphs, MATCH Commun. Math. Com- put. Chem. 64, 619–630, 2010.
  • [11] K.C. Das, I. Gutman and B. Furtula, On first geometric-arithmetic index of graphs, Discrete Appl. Math. 159, 2030–2037, 2011.
  • [12] K.C. Das, I. Gutman and B. Furtula, Survey on Geometric-Arithmetic Indices of Graphs, MATCH Commun. Math. Comput. Chem. 65, 595–644, 2011.
  • [13] S.S. Dragomir, A Survey On Cauchy-Bunyakovosky-Schwarz Type Discrete Inequali- ties, J. Inequal. Pure and Appl. Math. 4 (3), Art. 63, 2003.
  • [14] M. Drmota, Random Trees: An Interplay Between Combinatorics and Probability, Springer, Wien-New York, 2009.
  • [15] M. Eliasi, A. Iranmanesh and I. Gutman, Multiplicative versions of first Zagreb index, Commun. Math. Comput. Chem. 68 (1), 217–230, 2012.
  • [16] G.H. Fath-Tabar, Old and New Zagreb Indices of Graphs, MATCH Commun. Math. Comput. Chem. 65, 79–84, 2011.
  • [17] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (4), 1184–1190, 2015.
  • [18] X. Guo and Y. Gao Arithmetic-geometric spectral radius and energy of graphs, MATCH Commun. Math. Comput. Chem. 83, 651–660, 2020.
  • [19] I. Gutman, Degree–based topological indices, Croat. Chem. Acta 86, 351–361, 2013.
  • [20] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 83–92, 2004.
  • [21] I. Gutman and T. Réti, Zagreb group indices and beyond, Int. J. Chem. Model. 6 (2-3), 191–200, 2014.
  • [22] I. Gutman and J. Tošović, Testing the quality of molecular structure descriptors. Vertex–degreebased topological indices, J. Serb. Chem. Soc. 78 (6), 805–810, 2013.
  • [23] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535–538, 1972.
  • [24] J.C. Hernandez, J.M. Rodriguez and J.M. Sigarreta On the geometric-arithmetic in- dex by decompositions, J. Math. Chem. 55, 1376–1391, 2017.
  • [25] H. Kober, On the arithmetic and geometric means and on Hölderís inequality, Proc. Amer. Math. Soc. 9, 452–459, 1958.
  • [26] X. Li and H. Zhao, Trees with the first smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50, 57–62, 2004.
  • [27] X. Li and J. Zheng, A unified approach to the extremal trees for different indices, MATCH Commun. Math. Comput. Chem. 54, 195–208, 2005.
  • [28] M. Liu and B. Liu, Some properties of the first general Zagreb index, Australas. J. Combin. 47, 285–294, 2010.
  • [29] A. Martínez-Pérez, J.M. Rodríguez and J.M. Sigarreta, A new approximation to the geometric-arithmetic index, J. Math. Chem. 56, 1865–1883, 2018, DOI: 10.1007/s10910-017-0811-3.
  • [30] A. Miličević and S. Nikolić, On variable Zagreb indices, Croat. Chem. Acta 77, 97– 101, 2004.
  • [31] S. Nikolić, A. Miličević, N. Trinajstić and A. Jurić, On Use of the Variable Zagreb $^\nu M_2$ Index in QSPR: Boiling Points of Benzenoid Hydrocarbons, Molecules 9, 1208–1221, 2004.
  • [32] M. Randić, Novel graph theoretical approach to heteroatoms in QSAR, Chemometrics Intel. Lab. Syst. 10, 213–227, 1991.
  • [33] M. Randić, On computation of optimal parameters for multivariate analysis of structure-property relationship, J. Chem. Inf. Comput. Sci. 31, 970–980, 1991.
  • [34] M. Randić, D. Plavšić and N. Lerš, Variable connectivity index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 41, 657–662, 2001.
  • [35] J.M. Rodríguez, J.L. Sánchez and J.M. Sigarreta, On the first general Zagreb index, J. Math. Chem. 56, 1849–1864, 2018, DOI: 10.1007/s10910-017-0816-y.
  • [36] J.M. Rodríguez and J.M. Sigarreta, On the Geometric-Arithmetic Index, MATCH Commun. Math. Comput. Chem. 74, 103–120, 2015.
  • [37] J.M. Rodríguez and J.M. Sigarreta, Spectral properties of geometric-arithmetic index, Appl. Math. Comput. 277, 142–153, 2016.
  • [38] J.M. Rodríguez and J.M. Sigarreta, New Results on the Harmonic Index and Its Generalizations, MATCH Commun. Math. Comput. Chem. 78 (2), 387–404, 2017.
  • [39] J.M. Sigarreta, Bounds for the geometric-arithmetic index of a graph, Miskolc Math. Notes, 16 (2), 1199–1212, 2015.
  • [40] M. Singh, K.Ch. Das, S. Gupta and A.K. Madan, Refined variable Zagreb indices: highly discriminating topological descriptors for QSAR/QSPR, Int. J. Chem. Model- ing, 6 (2-3), 403–428, 2014.
  • [41] TRC Thermodynamic Tables. Hydrocarbons; Thermodynamic Research Center, The Texas A & M University System: College Station, TX, 1987.
  • [42] R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2), 359–372, 2010.
  • [43] M. Vöge, A.J. Guttmann and I. Jensen, On the number of benzenoid hydrocarbons, J. Chem. Inf. Comput. Sci. 42, 456–466, 2002.
  • [44] D. Vukičević, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta, 83, 261–273, 2010.
  • [45] D. Vukičević and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46, 1369–1376, 2009.
  • [46] D. Vukičević and M. Gašperov, Bond Additive Modeling 1. Adriatic Indices, Croat. Chem. Acta, 83 (3), 243–260, 2010.
  • [47] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69, 17–20, 1947.
  • [48] S. Zhang, W. Wang and T.C.E. Cheng, Bicyclic graphs with the first three smallest and largest values of the first general Zagreb index, MATCH Commun. Math. Comput. Chem. 55, 579–592, 2006.
  • [49] H. Zhang and S. Zhang, Unicyclic graphs with the first three smallest and largest values of the first general Zagreb index, MATCH Commun. Math. Comput. Chem. 55, 427–438, 2006.
  • [50] B. Zhou, I. Gutman and T. Aleksić, A note on Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 60, 441–446, 2008.
  • [51] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47, 210–218, 2010.
There are 51 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ana Granados 0000-0001-9775-8659

Ana Portilla 0000-0001-8008-8505

Jose Manuel Rodrıguez Garcıa 0000-0003-2851-7442

Jose Sigarreta This is me 0000-0003-4352-5109

Project Number MTM2016-78227-C2-1-P; MTM2017-90584-REDT
Publication Date June 7, 2021
Published in Issue Year 2021 Volume: 50 Issue: 3

Cite

APA Granados, A., Portilla, A., Rodrıguez Garcıa, J. M., Sigarreta, J. (2021). Inequalities on the geometric-arithmetic index. Hacettepe Journal of Mathematics and Statistics, 50(3), 778-794. https://doi.org/10.15672/hujms.749744
AMA Granados A, Portilla A, Rodrıguez Garcıa JM, Sigarreta J. Inequalities on the geometric-arithmetic index. Hacettepe Journal of Mathematics and Statistics. June 2021;50(3):778-794. doi:10.15672/hujms.749744
Chicago Granados, Ana, Ana Portilla, Jose Manuel Rodrıguez Garcıa, and Jose Sigarreta. “Inequalities on the Geometric-Arithmetic Index”. Hacettepe Journal of Mathematics and Statistics 50, no. 3 (June 2021): 778-94. https://doi.org/10.15672/hujms.749744.
EndNote Granados A, Portilla A, Rodrıguez Garcıa JM, Sigarreta J (June 1, 2021) Inequalities on the geometric-arithmetic index. Hacettepe Journal of Mathematics and Statistics 50 3 778–794.
IEEE A. Granados, A. Portilla, J. M. Rodrıguez Garcıa, and J. Sigarreta, “Inequalities on the geometric-arithmetic index”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, pp. 778–794, 2021, doi: 10.15672/hujms.749744.
ISNAD Granados, Ana et al. “Inequalities on the Geometric-Arithmetic Index”. Hacettepe Journal of Mathematics and Statistics 50/3 (June 2021), 778-794. https://doi.org/10.15672/hujms.749744.
JAMA Granados A, Portilla A, Rodrıguez Garcıa JM, Sigarreta J. Inequalities on the geometric-arithmetic index. Hacettepe Journal of Mathematics and Statistics. 2021;50:778–794.
MLA Granados, Ana et al. “Inequalities on the Geometric-Arithmetic Index”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, 2021, pp. 778-94, doi:10.15672/hujms.749744.
Vancouver Granados A, Portilla A, Rodrıguez Garcıa JM, Sigarreta J. Inequalities on the geometric-arithmetic index. Hacettepe Journal of Mathematics and Statistics. 2021;50(3):778-94.

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